FEROS Exposure Time Calculator

Created on September 17, 2019

2019   ·   astronomy   telescopes   python   ·   notes

Sources: University of Sheffield, ESO-ETC.

Astronomy and its tools

Astronomy is done through observing stars, galaxies, or other regions of space from various telescopes and through different instruments. One common instrument used in exoplanetary science are echelle spectrographs — and one such instrument is FEROS, mounted on the MPG/ESO 2.2-m telescope at La Silla Observatory, Chile. That’s the instrument for which we’ll build an Exposure Time Calculator (ETC).

What is an Exposure Time Calculator?

When observing through a telescope one does not blindly point at a star for an arbitrary amount of time. Instead, the exposure time is estimated to reach a target signal-to-noise ratio. The signal-to-noise ratio compares signal level to background noise: the higher the ratio, the more signal is in your data.

Understanding photons

The signal we collect from stars comes in the form of photons, so understanding how much signal we get means understanding how many photons we can detect.

In statistics, the probability of a given number of events occurring in a fixed interval of time — for example how many photons hit a telescope’s detector — is described by the Poisson distribution:

\[P(k) = e^{-\lambda}\frac{\lambda^{k}}{k!}\]

where \(\lambda\) is the event rate and \(k\) the number of events. An important property of the Poisson distribution is that its mean and standard deviation are \(\lambda\) and \(\sqrt{\lambda}\) respectively. In the limit of large \(k\) the Poisson distribution tends to a Normal distribution with mean and variance both equal to \(\lambda\).

Charge-Coupled Device

Another important component of telescopes is the Charge-Coupled Device (CCD) — the “camera” where photons land. When a photon hits a CCD pixel with enough energy, it excites and frees electrons. After an integration time \(t\), all pixels are shifted toward a final pixel where the electronics read them out as numbers.

Two CCD-specific noise sources matter here:

  1. Dark current — electrons freed by the intrinsic thermal activity of the CCD. The longer the CCD runs, the more of these accumulate.
  2. Read-out noise — time-independent noise added by the CCD while reading out.

The equation

We have four sources of noise:

  • Noise from the source object
  • Noise from the sky background
  • Noise from the dark current
  • Read-out noise

All but the read-out noise are Poissonian. If the number of electrons freed by the object, sky, and dark current are \(S_o\), \(S_b\), and \(S_d\), then their individual noises are \(\sqrt{S_o}\), \(\sqrt{S_b}\), and \(\sqrt{S_d}\). Assuming independence and the large-\(k\) Gaussian limit, variances add, so with read-out noise \(R\):

\[N = \sqrt{S_o + S_b + S_d + R^{2}}\]

The signal-to-noise ratio is then

\[S_o/N = \frac{S_o}{\sqrt{S_o + S_b + S_d + R^{2}}}.\]

In practice

Dark current and read-out noise are given by the instrument specs. Source and sky electron counts we compute ourselves.

Consider a star with visual magnitude \(m_V\), sky brightness \(m_{sky}\) in the V band, total efficiency \(\epsilon\), telescope mirror area \(A\), \(n_{pix}\) integration pixels, and spectral bin \(\Delta_S\) (wavelength range per pixel).

First correct the magnitude for airmass \(X\) and extinction \(k\):

\[m_{Vc} = m_V + k \cdot X.\]

Then the flux:

\[F = 10^{-0.4\,m_{Vc}} \cdot f_{0}\]

with \(f_0 = 3.60994 \times 10^{-9}\;\text{erg}/\text{cm}^{2}/\text{s}/\text{Å}\) for the V band.

The energy carried by a single photon is

\[E = \frac{h\,c}{\lambda},\]

so the electrons freed by the source are

\[S = \frac{F \cdot \Delta_S \cdot \epsilon \cdot A}{E}.\]

The sky calculation is identical. Over integration time \(t\) the counts scale as \(S \cdot t\).

Putting it together:

\[S/N = \frac{S_o \cdot t}{\sqrt{S_o\,t + S_b\,t\,n_{pix} + S_d\,t\,n_{pix} + R^{2}\,n_{pix}}}.\]

Results

Plugging in the MPG/ESO 2.2-m + FEROS numbers with an assumed extinction of 0.1 and a full-moon sky, below are signal-to-noise ratios for exposure times 10–60 min, magnitudes 6–20, and airmasses 1.0–2.5.

Code for reproducing these plots is available on GitHub.